Nonlinear, Data-based Reduced Models of Climate

Variability

Michael Ghil

Ecole Normale Supérieure, Paris, and University of California, Los Angeles

Joint work with Dmitri Kondrashov, UCLA; Sergey Kravtsov, U. Wisconsin–Milwaukee;Andrew Robertson, IRI, Columbiahttp://www.atmos.ucla.edu/tcd/

Motivation• Sometimes we have data but no models.• Linear inverse models (LIM) are good least-square fits to data,

but don’t capture all the processes of interest.• Difficult to separate between the slow and fast dynamics (MTV).• We want models that are as simple as possible, but not any

simpler.

Criteria for a good data-derived model

• Fit the data, as well or better than LIM.• Capture interesting dynamics: regimes, nonlinear oscillations.• Intermediate-order deterministic dynamics.• Good noise estimates.

Key ideas

• Nonlinear dynamics:

• Discretized, quadratic:

• Multi-level modeling of red noise:

Empirical mode reduction (EMR)–I

• Multiple predictors: Construct the reduced model

using J leading PCs of the field(s) of interest.

• Response variables: one-step time differences of predictors;

step = sampling interval = t.

• Each response variable is fitted by an independent

multi-level model:

The main level l = 0 is polynomial in the predictors;

all the other levels are linear.

Empirical mode reduct’n (EMR) – II• The number L of levels is such that each of the

last-level residuals (for each channel corresponding

to a given response variable) is “white” in time.

• Spatial (cross-channel) correlations of the last-level

residuals are retained in subsequent

regression-model simulations.

• The number J of PCs is chosen so as to optimize the

model’s performance.

• Regularization is used at the main (nonlinear) level

of each channel.

Illustrative example: Triple well

• V (x1,x2) is not polynomial!

• Our polynomial regression

model produces a time

series whose statistics

are nearly identical to

those of the full model!!

• Optimal order is m = 3;

regularization required

for polynomial models of

order m ≥ 5.

NH LFV in QG3 Model – IThe QG3 model (Marshall and Molteni, JAS, 1993):

• Global QG, T21, 3 levels, with topography;

perpetual-winter forcing; ~1500 degrees of freedom.

• Reasonably realistic NH climate and LFV:

(i) multiple planetary-flow regimes; and

(ii) low-frequency oscillations

(submonthly-to-intraseasonal).

• Extensively studied: A popular “numerical-laboratory” tool

to test various ideas and techniques for NH LFV.

NH LFV in QG3 Model – II

Output: daily streamfunction () fields ( 105 days)

Regression model:

• 15 variables, 3 levels (L = 3), quadratic at the main level

• Variables: Leading PCs of the middle-level

• No. of degrees of freedom = 45 (a factor of 40 less than

in the QG3 model)

• Number of regression coefficients P =

(15+1+15•16/2+30+45)•15 = 3165 (<< 105)

• Regularization via PLS applied at the main level.

NH LFV in QG3 Model – III

NH LFV in QG3 Model – IV

The correlation between the QG3 map and the EMR model’s map exceeds 0.9 for each cluster centroid.

NH LFV in QG3 Model – V

• Multi-channel SSA (M-SSA)

identifies 2 oscillatory

signals, with periods of

37 and 20 days.

• Composite maps of these

oscillations are computed

by identifying 8 phase categories, according to M-SSA reconstruction.

NH LFV in QG3 Model – VIComposite 37-day cycle:

QG3 and EMR results are virtually identical.

NH LFV in QG3 Model – VII

Regimes vs. Oscillations:

• Fraction of regime days as a function of

oscillation phase.

• Phase speed in the (RC vs. ∆RC) plane –

both RC and ∆RC are normalized so that

a linear, sinusoidal oscillation

would have a constant phase speed.

NH LFV in QG3 Model – VIIIRegimes vs. Oscillations:

• Fraction of

regime days:

NAO– (squares),

NAO+ (circles),

AO+ (diamonds);

AO– (triangles).

• Phase speed

NH LFV in QG3 Model – IX

• Regimes AO+, NAO– and NAO+ are associated with

anomalous slow-down of the 37-day oscillation’s

trajectory nonlinear mechanism.

• AO– is a stand-alone regime, not associated

with the 37- or 20-day oscillations.

Regimes vs. Oscillations:

NH LFV in QG3 Model – X

Quasi-stationary states

of the EMR model’s

deterministic

component.

Tendency threshold

a) = 10–6; and

) = 10–5.

NH LFV in QG3 Model – XI

37-day eigenmode

of the regression

model linearized

about climatology*

* Very similar to the composite 37-day oscillation.

NH LFV in QG3 Model – XII

Panels (a)–(d): noise amplitude = 0.2, 0.4, 0.6, 1.0.

Conclusions on QG3 Model• Our ERM is based on 15 EOFs of the QG3 model and has

L = 3 regression levels, i.e., a total of 45 predictors (*).

• The dynamical analysis of the reduced model

identifies AO– as the model’s unique steady state.

• The 37-day mode is associated, in the reduced model,

with the least-damped linear eigenmode.

(*) An ERM model with 4*3 = 12 variables only does not work!

• The ERM approximates the QG3 model’s major

statistical features (PDFs, spectra, regimes,

transition matrices, etc.) strikingly well.

• The additive noise interacts with the nonlinear dynamics to

yield the full ERM’s (and QG3’s) phase-space PDF.

NH LFV – Observed Heights

• 44 years of daily

700-mb-height winter data

• 12-variable, 2-level model

works OK, but dynamical

operator has unstable

directions: “sanity checks”

required.

Concluding Remarks – I• The generalized least-squares approach is well suited to

derive nonlinear, reduced models (EMR models) of

geophysical data sets; regularization techniques such as

PCR and PLS are important ingredients to make it work.

• The multi-level structure is convenient to implement and

provides a framework for dynamical interpretation

in terms of the “eddy–mean flow” feedback (not shown).

• Easy add-ons, such as seasonal cycle (for ENSO, etc.).

• The dynamic analysis of EMR models provides conceptual

insight into the mechanisms of the observed statistics.

Concluding Remarks – II

Possible pitfalls:

• The EMR models are maps: need to have an idea about

(time & space) scales in the system and sample accordingly.

• Our EMRs are parametric: functional form is pre-specified,

but it can be optimized within a given class of models.

• Choice of predictors is subjective, to some extent, but their

number can be optimized.

• Quadratic invariants are not preserved (or guaranteed) –

spurious nonlinear instabilities may arise.

ReferencesKravtsov, S., D. Kondrashov, and M. Ghil, 2005:

Multilevel regression modeling of nonlinear processes:

Derivation and applications to climatic variability.

J. Climate, 18, 4404–4424.

Kondrashov, D., S. Kravtsov, A. W. Robertson, and M. Ghil, 2005: A hierarchy of data-based ENSO models.

J. Climate, 18, 4425–4444.

Kondrashov, D., S. Kravtsov, and M. Ghil, 2006:

Empirical mode reduction in a model of extratropical

low-frequency variability. J. Atmos. Sci., accepted.

http://www.atmos.ucla.edu/tcd/

NomenclatureResponse variables:

Predictor variables:

• Each is normally distributed about

• Each is known exactly. Parameter set {ap}:

– known dependence

of f on {x(n)} and {ap}.

REGRESSION: Find

LIM extension #1• Do a least-square fit to a nonlinear function of the data:

J response variables:

Predictor variables (example – quadratic polynomial

of J original predictors):

Note: need to find many more regression coefficients than

for LIM; in the example above P = J + J(J+1)/2 + 1 = O(J2).

Regularization

• Regularization involves rotated predictor variables:

the orthogonal transformation looks for an “optimal”

linear combination of variables.

• “Optimal” = (i) rotated predictors are nearly uncorrelated; and

(ii) they are maximally correlated with the response.

• Canned packages available.

• Caveat: If the number P of regression parameters is

comparable to (i.e., it is not much smaller than) the

number of data points, then the least-squares problem may

become ill-posed and lead to unstable results (overfitting) ==>

One needs to transform the predictor variables to regularize

the regression procedure.

LIM extension #2

Main level, l = 0:

Level l = 1:

… and so on …

Level L:

• rL – Gaussian random deviate with appropriate variance

• If we suppress the dependence on x in levels l = 1, 2,… L,

then the model above is formally identical to an ARMA model.

• Motivation: Serial correlations in the residual.